How Does the Limit Process Simplify the Expression x^2-36?

[fermio]

Homework Statement

\lim_{x\to 6}\frac{\sqrt{x+3}-3}{x-6}

Homework Equations

Answer is 1/6.

The Attempt at a Solution

\lim_{x\to 6}\frac{\sqrt{x+3}-3}{x-6}=\lim_{x\to 6}\frac{(x+3-9)(x+6)}{(x^2-36)(\sqrt{x+3}+3)}

 

[Mathdope]

Homework Statement

\lim_{x\to 6}\frac{\sqrt{x+3}-3}{x-6}

Homework Equations

Answer is 1/6.

The Attempt at a Solution

\lim_{x\to 6}\frac{\sqrt{x+3}-3}{x-6}=\lim_{x\to 6}\frac{(x+3-9)(x+6)}{(x^2-36)(\sqrt{x+3}+3)}

Look more carefully at your attempt at a solution. You actually have it.

 

[arildno]

Do not bother with expanding with (x+6) as well:
\frac{\sqrt{x+3}-3}{(x-6)}=\frac{x+3-9}{(x-6)(\sqrt{x+3}+3)}=\frac{(x-6)}{(x-6)(\sqrt{x+3}+3)}

 

[fermio]

"Look more carefully at your attempt at a solution. You actually have it."
I don't see.
\lim_{x\to 6}\frac{\sqrt{x+3}-3}{x-6}=\lim_{x\to 6}\frac{(x+3-9)(x+6)}{(x^2-36)(\sqrt{x+3}+3)}=\lim_{x\to 6}\frac{x^2+6x-6x-36}{x^2\sqrt{x+3}+3x^2-36\sqrt{x+3}-108}

I get it.
\lim_{x\to 6}\frac{\sqrt{x+3}-3}{x-6}=\lim_{x\to 6}\frac{x+3-9}{(x-6)(\sqrt{x+3}+3)}=\lim_{x\to 6}\frac{1}{\sqrt{x+3}+3}=\frac{1}{6}

 

[Rainbow Child]

How can you write x^2-36?